Problem 42 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\alpha, \beta, \gamma\) are the roots of \(x^{3}+p x^{2}+2 x+p=0\), the general value of \(\tan ^{-1} \alpha+\tan ^{-1} \beta+\tan ^{-1} \gamma\) is (a) \(n \pi\) (b) \(n \pi / 2\) (c) \((2 n+1) \pi / 2\) (d) depend on \(p\).

Step-by-Step Solution

Answer

The general value of \(\tan^{-1} \alpha + \tan^{-1} \beta + \tan^{-1} \gamma\) will depend on \( p \), therefore option (d) is correct.

Step 1: Rearrangement and applying the formula of the sum of roots

The sum of the roots of the given cubic equation is \(-p\). So we have \(\alpha + \beta + \gamma = -p\). Taking arctangent on both sides we can write \(\tan^{-1} \alpha + \tan^{-1} \beta + \tan^{-1} \gamma = \tan^{-1}(-p)\).

Step 2: Simplifying using the arctangent sum formula

From the beginners guide to trigonometry, the sum formula for the arctangent function of two angles \(a\) and \(b\) is given as: \(\tan^{-1} a + \tan^{-1} b = \tan^{-1} ((a + b) / (1 - a*b))\) unless \(a*b > 1\). In which case the sum is given by \(\tan^{-1} ((a + b) / (1 - a*b)) + \pi\). Simultaneously applying this formula for \(\tan^{-1} \alpha + \tan^{-1} \beta + \tan^{-1} \gamma\), the goal would be to simplify to \(\tan^{-1} (-p)\).

Step 3: Determine the conditions under which the equation is true

In order to determine the relation between \( n \pi \), \( n \pi / 2 \), \( (2 n + 1) \pi / 2 \) and \( p \), we will substitute each of the values for each option to find the one that makes our equation true.

Key Concepts

Sum of Roots

In a cubic equation like \(x^3 + px^2 + 2x + p = 0\), the sum of the roots \(\alpha, \beta,\) and \(\gamma\) can be calculated using Vieta’s formulas. Let's break this down! Vieta’s formulas state that the sum of the roots, for a polynomial equation of the form \(ax^3 + bx^2 + cx + d = 0\), is given by \(-\frac{b}{a}\). Here, since \(a = 1\) and \(b = p\), the sum of the roots becomes \(-p\). This means that:

• \(\alpha + \beta + \gamma = -p\)

Now, even though finding the actual numeric value of \(\alpha, \beta,\) and \(\gamma\) might not always be necessary, understanding their sum is crucial in evaluating trigonometric expressions involving these roots. Vieta's formulas give us a neat way to summarize the roots without needing their specific values.
Remember, this can be a powerful tool when you tackle quadratic, cubic, or any polynomial equation, allowing you to grasp more complex algebraic sums and relations easily.

Arctangent Function

The arctangent function, also known as the inverse tangent function, is denoted by \(\tan^{-1} x\). This function is crucial when dealing with trigonometric identities involving angles. The expression \(\tan^{-1} \alpha + \tan^{-1} \beta + \tan^{-1} \gamma\) refers to the combined angle of the roots taken in inverse tangent form.
To simplify expressions of this type, we often use the arctangent sum formula:

• \( \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \)\
• If \(ab > 1\), then we add \( \pi \) to the result.

For the given exercise, the roots' values are plugged into this identity to reduce the sum to a form involving only \(p\). This approach helps to combine multiple inverse tangent terms into a single, more easily calculable arctangent.
Understanding this function's role and manipulation is fundamental in problems where trigonometric identities are used to simplify angles and expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true. When dealing with the sum of inverse tangent expressions like \(\tan^{-1} \alpha + \tan^{-1} \beta + \tan^{-1} \gamma\), these identities become powerful tools to simplify complex expressions.
One commonly used identity is the arctangent sum formula. This identity helps you transform a complex angle sum into a simpler form:

• \(\tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right)\),
• possibly adding \(\pi\) if \(ab > 1\).

Trigonometric identities often allow for the transformation of expressions into forms where they either become zero or a simple multiple of \(\pi\). This simplification provides insight into the relations between trigonometric expressions and polynomial roots.
In practical applications, especially in exams or problem-solving, mastering these identities provides a quick path to solutions that might otherwise take much longer to compute or guess.